I need A neighbourhood to listen to my authority. I do this by killing Bob. I could buy a knife and stab Bob ($5), OR I could buy my very own M1 Abrams tank (a ******* lot of money) and blow his whole ******* house up. He's just as dead with a knife wound, but the rest of the neighbourhood is way more scared of a goddamn tank.

Intimidation factor and there is a throwaway line by Han when they reach the remnants of Aldaraan about how it would take the fire power of half the star fleet or something along those lines. My memory of it is hazy and I can't find a source for it at the moment.

In the expanded universe, it was partially built, but destroyed before it was completed. As far as I remember, and fourth one was built, but on a much smaller scale. It had none of the weaknesses of the predecessors, but a small ship rammed it, destroying it. It was either that, or the ship fired one of its supernova missiles (seriously, you shoot a star with it, and the star goes supernova). The ship could withstand ramming pretty much anything, and it was used as a fairly effective weapon against capital ships.

You are thinking of the Sun Crusher and the Prototype Death Star. The supernova torpedoes did do some damage to the Death Star but not enough to be really effective. Ultimately the pilot of the Sun Crusher suckered the Death Star too close to a black hole and both were unable to escape it's gravitational pull. The pilot of the Sun Crusher survived by breaking his legs and cramming himself into a message capsule.